Linear Algebra with NumPy Primer Flashcards Preview

11637 Foundations of Computational Data Science > Linear Algebra with NumPy Primer > Flashcards

Flashcards in Linear Algebra with NumPy Primer Deck (52)
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1

What does this mean?

the set of all real-valued matrices with n rows and d columns

2

What is this called? What does it mean?

1(x)

  • binary indicator function,
  • returns 1 if x is a true boolean and 0 otherwise

3

What does this mean?

the set of all real-valued n-dimensional vectors

4

What does this mean?

the set of real numbers

5

What is this?

  • A pair of input vector and output scalar
  • The superscript indicates the index of this pair, not the exponent. In other words, x(5) is the input vector at index 5, not x raised to the power of 5.

6

What is this?

|C|

  • Determinant if C is a matrix
  • Cardinality (number of member elements) if C is a set.

7

What's a column vector?

Each entry occupies one row.

8

What's this?

One use: denotes the i-th entry of a column vector

9

What does this mean?

Row vector. Each entry covers 1 column

10

For matrix A, what is 

denotes the i-th row of A

11

For matrix A, what is 

denotes the i-th column of A

12

What can dot product be applied to? What's the formula?

Two vectors of same dimensions

13

What's the inner product?

Same as dot product

14

  • What can outer product be applied to?
  • What does it result in?
  • What's the formula?

  • Vectors (can have different number of elements)

15

What's another way to express a dot product for two column vectors x and y?

16

What's this for two vectors x and y?

The outer product

17

What's a matrix-vector product?

18

  • What are the properties of matrix multiplication? (2)
  • What properties does it not have? (1)

Matrix multiplication is:

  • associative, (AB)C=A(BC)
  • distributive, A(B+C)=AB+AC

Is not:

  • commutative, AB≠BA

19

What's a property of transposes?

20

What's the Hadamard product?

elementwise matrix multiplication

21

What does this mean?

A standard basis vector - ith entry is 1 and the other entries are all 0

22

What does this denote?

Identity matrix of nxn size

23

What's the point of a basis vector?

By multiplying a basis vector with a vector x∈Rn, we obtain the single entry xi

24

Describe the idea of one-hot encoding

  • In each row of the resulting one-hot-encoded matrix:
  • In each row of the original vector, find the value
    • This value will serve as the column index of the "1" entry in the same row in the corresponding matrix. 
    • The remaining values in that row are 0

25

What's a symmetric matrix? 

26

What's this? What can it be applied to?

  • Inverse matrix
  • Only applied to square matrices

27

What's the fundamental rule of matrix inverses?

28

What are the properties of matrix inverses?

29

What's a diagonal matrix?

a rectangular matrix with non-zero entries only on the main diagonal

30

What's this? 

  • The p-norm (a vector norm). 
  • It's indicative of its magnitude or length